# Internal division

Coordinates of the point C which divides the line segment joining the points A(x1, y1) and B(x2, y2) internally in the ratio m : n are given by

x =

# External division

Coordinates of the point C which divides the line segment joining the points A(x1, y1) and B(x2, y2) externally in the ratio m : n are given by

x =

Notes:
• The midpoint of (x1y1) and (x2y2) is .
• The centroid (G) (concurrency point of medians of a triangle) of a triangle having vertices (x1y1), (x2y2), and (x3,y3) is G  using the fact that centroid is concurrency point of medians, and G divides median AD in ratio 2:1.
• Orthocenter of a triangle is point of concurrency of altitudes of triangle.
• In a triangle (not equilateral) orthocenter (H) (concurrency point of altitudes of triangle), centroid (G), and circumcenter (O) are collinear and in the ratio HG/GO= 2.
• If circumcenter of a triangle (concurrency point of perpendicular bisector of sides of triangle) with vertices (x1y1), (x2y2), and (x3y3) is origin (0, 0) then orthocenter is (x1 + x2 + x3y1 + y2 +y3).
• Incenter of a triangle (concurrency point of internal angle bisector) with verticesA(x1y1), B(x2y2), and C(x3y3) is I using thefact that internal angle bisector ADdivides BC in the ratio AB/AC.
Excenters

Let A(x1y1), B(x2y2), and C(x3y3) be the vertices of the triangle ABC, and let ab, andc be the lengths of the sides BCCA, and ABrespectively.

The circle which touches the sides BC and two sides AB and AC produced is called the escribed circle opposite to the angle A.

The bisectors of the external angle B andC meet at a point I1 which is the center of the escribed circle opposite to the angle A.

The coordinates of I1 are given by

The coordinates of I2 and I3 (centers of escribed circles opposite to the angles B and C respectively) are given by

and
• To prove that ABCD are vertices of
 Parallelogram Show that diagonals AC and BD bisect each other. Rhombus Show that diagonals AC and BD bisect each other and adjacent sides AB and BC are equal. Square Show that diagonals AC and BD equal and bisect each other and adjacent sides AB and BC are equal. Rectangle Show that diagonals AC and BD equal and bisect each other.